The Australian parliament must commit to deliver marriage equality by Christmas, the prime minister, Malcolm Turnbull, has said after an “unequivocal, overwhelming” vote of 61.6% in favour of same-sex marriage in an unprecedented national postal survey.

As nationwide celebrations heralded a result that will give enormous momentum to a final push to achieve the historic social reform, Turnbull moved to head off attempts from conservatives in his ruling Liberal-National Coalition to frustrate or delay the legislative process.

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Turnbull said the survey – which had a participation rate of 79.5% – meant Australians had “spoken in their millions and they have voted overwhelmingly yes for marriage equality”.

“They voted yes for fairness, yes for commitment, yes for love. And now it is up to us here in the parliament of Australia to get on with it, to get on with the job the Australian people asked us to do and get this done,” he said.

After a national vote that was resisted at every turn by marriage equality advocates who viewed it as an affront because it determined the right to equality before the law by a majoritarian vote, prominent LGBTI Australians celebrated that the Australian values of fairness and equality were reflected in the outcome.

In Melbourne 5,000 people outside the State Library of Victoria cheered and danced to Kylie Minogue. In Sydney’s Prince Alfred Park, John Paul Young sang Love Is in the Air.

The swimming champion Ian Thorpe, the actor Magda Szubanski, the Qantas chief executive, Alan Joyce, and the Labor leader in the Senate, Penny Wong, were among those overcome with emotion and keen to share their joy with fellow Australians. Celebrities including Minogue, Ellen DeGeneres – married to Australian-American Portia de Rossi – the Apple boss, Tim Cook, the Canadian leader, Justin Trudeau, and former British prime minister David Cameron tweeted congratulations.

For Hannah Collins and Heather Ford, there was no time to waste. When Kalisch finally announced that the yes campaign had secured 61.6% of the vote in Australia’s voluntary same-sex marriage postal survey, Ford proposed. “Heather got down on her knee, mate, and the date is 4 April,” Collins said.

Christine Forster, the sister of the former prime minister Tony Abbott – she is in a same-sex relationship and is a long-term supporter of marriage equality, while he is an opponent – punched the air. Szubanski said the result was for “all of us”.

“No matter how we want to live our life, we must live as equal people in this country,” she told the crowd in Sydney.

In Melbourne Andrew Doherty said he could finally plan the beach wedding he’s always dreamed of. He proposed to his partner of three years before the postal vote result announcement on Wednesday.

“I’m not happy we had to have this vote but I’m happy we have the opportunity to change things,” he said. “I’m confident Australia has woken up but I’m not confident the politicians have, especially given these bills proposed to embed discrimination.”

Every state and territory voted for marriage equality, with the national vote 7,817,247 in favour and 4,873,947 against. The constituencies of central Sydney and Melbourne saw the largest majorities in favour, at 83.7% each. Seventeen of the federal parliament’s 150 seats had a majority of no voters, including a conspicuous swath of western Sydney, a socially conservative and ethnically diverse area.

Crowds gathered in cities across Australia – at 7am in Perth, and in the rain in Adelaide. “Absolute elation” was reported in Brisbane.

At a press conference in Canberra, Wong said: “Thank you Australia, thank you for standing up for fairness, thank you for standing up for equality … for the LGBTI community everywhere … for the sort of Australia we believe in, that is decent, that is fair, which is accepting, which turns its back on exclusion and division.”

At a rally in Melbourne the opposition leader, Bill Shorten, promised: “Today we celebrate, tomorrow we legislate.

“It may have been 61% who voted yes in the survey, but I want to say to all LGBTQI Australians, you are 100% loved, 100% valued and, after the next two weeks of parliament, 100% able to marry the person you love,” he said.

In a speech after the result an Equality Campaign spokesman, Alex Greenwich, said: “Today love has had a landslide victory. Together we have achieved something truly remarkable, a win for fairness and equality, not only for the LGBTI community and our families, but for all Australians.”

Greenwich said the campaign had made more than 1m phone calls and knocked on 100,000 doors, an “unprecedented” level of support that had exceeded “any campaign in our history”.

“In doing so it has delivered an unequivocal mandate to federal parliament to vote this through by the end of the year,” he said.

Turnbull, same-sex marriage supporters in Australia’s ruling Coalition, the Labor opposition, the Greens and other crossbench parties have reached a consensus around a cross-party bill that makes minimalist changes to protect religious freedom without legalising discrimination by commercial service providers, such as cake makers – as some conservatives in the Coalition government have demanded.

On Wednesday afternoon the cross-party group capitalised on the mandate in the survey by introducing the bill and a motion that will lead to it being debated until it is passed in the sitting week beginning 27 November.

Conservatives including Abbott and the finance minister, Mathias Cormann, have foreshadowed that they will back amendments in parliament while other government MPs have said they will abstain if they don’t get their way on religious protections.

But the cross-party group has sufficient numbers to legislate marriage equality, provided that Turnbull’s position that parliament will determine which bill to use and amendments to make holds. The cross-party group were boosted when an alternative bill, by the Liberal senator James Paterson, was withdrawn, with Paterson conceding it had no chance of becoming law.

Same-sex marriage has been banned in Australia since 2004 when the Howard government changed the Marriage Act to define marriage as between a man and a woman. As many comparable countries such as the US and Britain allowed or legislated for same-sex marriage, Australia looked increasingly out of step.

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Turnbull pledges same-sex marriage will be law by Christmas – video

After the successful marriage equality referendum in Ireland in May 2015, pressure grew on the Australian government to legislate but the Coalition party room agreed on a national plebiscite instead.

The yes camp’s Equality Campaign combined with moderate Liberals, Labor, the Greens, unions and the progressive campaign organisation GetUp to argue that same-sex marriage was a matter of equality, fairness and allowing LGBTI Australians to marry the one they love.